What is the interval heights of 95% of women

Question

Assume the heights of women is approximately normal distributed, with the mean of 65.5 inches, and standard deviation of 2.5 inches. What is the interval of the heights of 95% of the women?

Answer 1

The area under the distribution curve between the blue lines is the required interval. The left blue line corresponds to Z=-1.96 and the right blue line to Z=1.96. Each of the areas in the tails sum to 5%, so because of symmetry, the area of the right tail is 2.5%. The Z value of 1.96 corresponds to 97.5%, so we need to find what H (high) and L (low) values this corresponds to:

1.96=(H-65.5)/2.5 and -1.96=(L-65.5)/2.5.

4.9=H-65.5, -4.9=L-65.5, H=70.4 in, L=60.6 in.

So the interval could be written 65.5±4.9, 60.6≤X≤70.4, or [60.6,70.4].